The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421, and is the same as the rectified 142.

For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure.[1] It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.

E. L. Elte named it V240 (for its 240 vertices) in his 1912 listing of semiregular polytopes.[2]

The 240 vertices of the 421 polytope can be constructed in two sets: 112 (22×8C2) with coordinates obtained from (±2,±2,0,0,0,0,0,0){\displaystyle (\pm 2,\pm 2,0,0,0,0,0,0)\,} by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from (±1,±1,±1,±1,±1,±1,±1,±1){\displaystyle (\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,} by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be a multiple of 4).

Each vertex has 56 nearest neighbors; for example, the nearest neighbors of the vertex (1,1,1,1,1,1,1,1){\displaystyle (1,1,1,1,1,1,1,1)} are those whose coordinates sum to 4, namely the 28 obtained by permuting the coordinates of (2,2,0,0,0,0,0,0){\displaystyle (2,2,0,0,0,0,0,0)\,} and the 28 obtained by permuting the coordinates of (1,1,1,1,1,1,−1,−1){\displaystyle (1,1,1,1,1,1,-1,-1)}. These 56 points are the vertices of a 321 polytope in 7 dimensions.

Each vertex has 126 second nearest neighbors: for example, the nearest neighbors of the vertex (1,1,1,1,1,1,1,1){\displaystyle (1,1,1,1,1,1,1,1)} are those whose coordinates sum to 0, namely the 56 obtained by permuting the coordinates of (2,−2,0,0,0,0,0,0){\displaystyle (2,-2,0,0,0,0,0,0)\,} and the 70 obtained by permuting the coordinates of (1,1,1,1,−1,−1,−1,−1){\displaystyle (1,1,1,1,-1,-1,-1,-1)}. These 126 points are the vertices of a 231 polytope in 7 dimensions.

Each vertex also has 56 third nearest neighbors, which are the negatives of its nearest neighbors, and one antipodal vertex, for a total of 1+56+126+56+1=240{\displaystyle 1+56+126+56+1=240} vertices.

Removing the node on the end of the 2-length branch leaves the 7-orthoplex in its alternated form (411):

Every 7-simplex facet touches only 7-orthoplex facets, while alternate facets of an orthoplex facet touch either a simplex or another orthoplex. There are 17,280 simplex facets and 2160 orthoplex facets.

Since every 7-simplex has 7 6-simplex facets, each incident to no other 6-simplex, the 421 polytope has 120,960 (7×17,280) 6-simplex faces that are facets of 7-simplexes. Since every 7-orthoplex has 128 (27) 6-simplex facets, half of which are not incident to 7-simplexes, the 421 polytope has 138,240 (26×2160) 6-simplex faces that are not facets of 7-simplexes. The 421 polytope thus has two kinds of 6-simplex faces, not interchanged by symmetries of this polytope. The total number of 6-simplex faces is 259200 (120,960+138,240).

The vertex figure of a single-ring polytope is obtained by removing the ringed node and ringing its neighbor(s). This makes the 321 polytope.

Mathematical representation of the physical Zome model isomorphic (?) to E8. This is constructed from VisibLie_E8 pictured with all 3360 edges of length √2(√5-1) from two concentric 600-cells (at the golden ratio) with orthogonal projections to perspective 3-space

The actual split real even E8 421 polytope projected into perspective 3-space pictured with all 6720 edges of length √2[5]

These graphs represent orthographic projections in the E8,E7,E6, and B8,D8,D7,D6,D5,D4,D3,A7,A5Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

The 421 polytope is last in a family called the k21 polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).

The 421 polytope can be projected into 3-space as a physical vertex-edge model. Pictured here as 2 concentric 600-cells (at the golden ratio) using Zome tools.[6] (Not all of the 3360 edges of length √2(√5-1) are represented.)

The 421 is related to the 600-cell by a geometric folding of the Coxeter-Dynkin diagrams. This can be seen in the E8/H4 Coxeter plane projections. The 240 vertices of the 421 polytope are projected into 4-space as two copies of the 120 vertices of the 600-cell, one copy smaller (scaled by the golden ratio) than the other with the same orientation. Seen as a 2D orthographic projection in the E8/H4 Coxeter plane, the 120 vertices of the 600-cell are projected in the same four rings as seen in the 421. The other 4 rings of the 421 graph also match a smaller copy of the four rings of the 600-cell.

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a rectification of the 421. Vertices are positioned at the midpoint of all the edges of 421, and new edges connecting them.

These graphs represent orthographic projections in the E8,E7,E6, and B8,D8,D7,D6,D5,D4,D3,A7,A5Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

These graphs represent orthographic projections in the E8,E7,E6, and B8,D8,D7,D6,D5,D4,D3,A7,A5Coxeter planes. Edges are not drawn. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green, etc.

These graphs represent orthographic projections in the E7,E6, and B8,D8,D7,D6,D5,D4,D3,A7,A5Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.